ON SOME PROPERTIES OF THE DIGIT SHIFT OPERATOR Q_s^*-REPRESENTATION OF REAL NUMBERS AND UNIFORMLY DISTRIBUTED SEQUENCES PRODUCED BY IT

Abstract

The work is devoted to the study of the properties of the left-shift operator $Q_{s}^{*}$-representation of real numbers and the study of the type of distribution of the sequences produced by the corresponding operator. The $Q_{s}^{*}$-representation of real numbers is a natural generalization of the classical s-representation and is topologically similar to the latter. E. Borel's classic result that almost all numbers are s-normal was over time translated into the terms of uniformly distributed sequences produced by the left-shift operator of the digits of the corresponding representation. It was proved that a number is s-normal only when the corresponding sequence generated by this number in the sense of the left shift operator is uniformly distributed. Despite the topological similarity between the $Q_{s}^{*}$-representation of real numbers and the classical s-representation, proving similar results for the former requires fundamentally new approaches that include the use of the apparatus of ergodic theory. The absence of the effect of metric transitivity of the appearance of digits, which is characteristic of the classical s-representation, does not allow the use of appropriate approaches to the $Q_{s}^{*}$-representation. The construction of normal numbers in various representation systems is a separate non-trivial problem and is the subject of many studies. In many cases, criteria for the normality of numbers, which can have a continuous structure (similar to the classical criteria of uniform distribution of the sequence) or a discrete structure, are useful for constructing the corresponding numbers. This paper presents generalizations of discrete criteria for the normality of numbers, which applied both to the classical s-representation and to the $Q_{s}$-representation of real numbers (the latter is a partial case of the $Q_{s}^{*}$-representation).